Let $S_n$ denote the set of all positive integers in the interval $[1,10^n]$ .

If four positive integers are taken at random from the set $S_n$ and multiplied together, find the probability, $P_n$ that the last digit is $1, 3 ,7,$ or $9$ .

Submit your answer as $\displaystyle \lim_{n\to\infty} P_n$ .

$\frac{1}{625}$
$\frac{1}{256}$
$\frac{16}{625}$
$\frac{8}{256}$

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The last digit of the four integers can only be 1,3,7,9(or it would be a mutiple of 2 or 5),so the probability of choosing one integer right is $\frac{4}{10}$ ,and choosing four right would be $(\frac{4}{10})^{4}=\frac{16}{625}$